Exercise 5.4.1 (Equation $x^2 + y^2 = 9z+3$)

Question

Show that the equation $x^2 + y^2 = 9z+3$ has no integral solution.

Richard Ganaye · Accepted Answer

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

\begin{proof} 
If reduced modulo $9$, this equation has no solution: for all integers $x,y$,
$$x^2 \equiv 0, 1, 4,7 \pmod 9,$$
thus
$$x^2 + y^2 \equiv 0, 1,2,4,5,7,8\pmod9.$$
Therefore the congruence
$$x^2 + y^2 \equiv 3 \pmod 9$$
has no solution, thus
$$x^2 + y^2 = 9z+3$$
has no integral solution.

\end{proof}
With Sagemath:
\begin{verbatim}
sage: a = set()
sage: for x in Integers(9):
....:     for y in Integers(9):
....:         a.add(x^2 + y^2)
....:         
sage: a
{0, 1, 2, 4, 5, 7, 8}
\end{verbatim}

Exercise 5.4.1 (Equation $x^2 + y^2 = 9z+3$)

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Exercise 5.4.1 (Equation $x^2 + y^2 = 9z+3$)

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